Geodesic
Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping
Electronic journal of differential equations, Tome 2002 (2002)
In this paper we obtain an exponential rate of decay for the solution of the viscoelastic nonlinear wave equation

$ u_{tt}-\Delta u+f(x,t,u)+\int_0^tg(t-\tau )\Delta u( \tau )\,d\tau +a(x)u_t=0\quad \hbox{in }\Omega\times (0,\infty ). $

Here the damping term $a(x)u_t$ may be null for some part of the domain $\Omega$. By assuming that the kernel $g$ in the memory term decays exponentially, the damping effect allows us to avoid compactness arguments and and to reduce number of the energy estimates considered in the prior literature. We construct a suitable Liapunov functional and make use of the perturbed energy method.
Classification : 35L05, 35L70, 35B40, 74D10
Keywords: semilinear wave equation, memory, localized damping 
@article{EJDE_2002__2002__a63,
     author = {Cavalcanti,  Marcelo M. and Domingos Cavalcanti,  Val\'eria N. and Soriano,  Juan A.},
     title = {Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping},
     journal = {Electronic journal of differential equations},
     year = {2002},
     volume = {2002},
     zbl = {0997.35037},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a63/}
}
TY  - JOUR
AU  - Cavalcanti,  Marcelo M.
AU  - Domingos Cavalcanti,  Valéria N.
AU  - Soriano,  Juan A.
TI  - Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping
JO  - Electronic journal of differential equations
PY  - 2002
VL  - 2002
UR  - http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a63/
LA  - en
ID  - EJDE_2002__2002__a63
ER  - 
%0 Journal Article
%A Cavalcanti,  Marcelo M.
%A Domingos Cavalcanti,  Valéria N.
%A Soriano,  Juan A.
%T Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping
%J Electronic journal of differential equations
%D 2002
%V 2002
%U http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a63/
%G en
%F EJDE_2002__2002__a63
Cavalcanti,  Marcelo M.; Domingos Cavalcanti,  Valéria N.; Soriano,  Juan A. Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electronic journal of differential equations, Tome 2002 (2002). http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a63/